Questions: In this problem, please evaluate the trig functions without a calculator and do not use a decimal point in your answer. An equation of the tangent line to the curve y=sin(x) at x=π / 2 is y=θ∇ ⋅(x-π / 2) An equation of the tangent line to the curve y=cos(x) at x=π / 4 is y=1/√2 +⋅(x-π / 4)

Solution Steps

To find the equation of the tangent line to a curve at a given point, we need to determine the slope of the tangent line at that point, which is given by the derivative of the function evaluated at that point. Then, we use the point-slope form of the equation of a line.

1. For the curve \( y = \sin(x) \) at \( x = \pi/2 \):

   • Find the derivative of \( y = \sin(x) \).
   • Evaluate the derivative at \( x = \pi/2 \) to get the slope.
   • Use the point-slope form of the line equation with the point \( (\pi/2, \sin(\pi/2)) \).

2. For the curve \( y = \cos(x) \) at \( x = \pi/4 \):

   • Find the derivative of \( y = \cos(x) \).
   • Evaluate the derivative at \( x = \pi/4 \) to get the slope.
   • Use the point-slope form of the line equation with the point \( (\pi/4, \cos(\pi/4)) \).

The derivative of \( y = \sin(x) \) is: \[ \frac{dy}{dx} = \cos(x) \]

\[ \left. \frac{dy}{dx} \right|_{x = \frac{\pi}{2}} = \cos\left(\frac{\pi}{2}\right) = 0 \]

\[ y\left(\frac{\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) = 1 \]

Using the point-slope form of the line equation: \[ y - y_1 = m(x - x_1) \] where \( m = 0 \), \( x_1 = \frac{\pi}{2} \), and \( y_1 = 1 \): \[ y - 1 = 0 \cdot (x - \frac{\pi}{2}) \] \[ y = 1 \]

The derivative of \( y = \cos(x) \) is: \[ \frac{dy}{dx} = -\sin(x) \]

\[ \left. \frac{dy}{dx} \right|_{x = \frac{\pi}{4}} = -\sin\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \]

\[ y\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]

Using the point-slope form of the line equation: \[ y - y_1 = m(x - x_1) \] where \( m = -\frac{\sqrt{2}}{2} \), \( x_1 = \frac{\pi}{4} \), and \( y_1 = \frac{\sqrt{2}}{2} \): \[ y - \frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{2}(x - \frac{\pi}{4}) \] \[ y = -\frac{\sqrt{2}}{2}(x - \frac{\pi}{4}) + \frac{\sqrt{2}}{2} \]

Final Answer